Case studies of real populations

Now that the theoretical foundations have been built to calculate the number of grains required for a statistically adequate provenance study, they will be tested on real data. Unfortunately, no population is ever completely known (certainly not if we consider that most published provenance studies work with fewer than 117 ages per sample). Therefore, two relatively large published detrital data sets will be used as a proxy for the populations that they were sampled from. By randomly selecting numbers from these "populations" with replacement, we can generate synthetic samples. This procedure is similar to what is called "bootstrapping" in the statistical literature [10].

Avigad et al. [11] published a set of 157 concordant single zircon U/Pb ages from the Early Paleozoic Nubian Sandstone. The vast majority of these grains are of Pan-African age (900-540 Ma) with relatively few older grains. Therefore, the population is relatively "easy" to sample (Figure 6). 1000 "bootstrap samples" of k numbers were selected from the data set for each value of M between 1 and 50, where the latter value is assumed to be the highest number of bins that one would ever want to use in a grain-age histogram. Similar to the algorithm that was used in the previous section, the proportion of the 1000 samples that miss at least one of the relevant fractions (f$\geq$0.05) was calculated. This exercise was done for k=60, k=95 and k=117 (Figure 6). The p$_{max}$ vs. M curve of this figure is much more irregular than that of Figures 4 and 1, because it represents only one, irregular population, and not the composite of a thousand random distributions (Figure 4), or one smooth uniform analytical distribution (Figure 1). For k=60, the maximum value for p is 8.8%. As expected, this is less than the 64% predicted by Equation 4, but almost twice as much as predicted by Equation 1. The fact that even samples from this "easy" population have $\geq$5% chance of missing at least one fraction $\geq$0.05 is another confirmation that 60 grains is not enough to attain the degree of adequacy many would consider necessary for a good provenance study. The maximum probability of missing any fraction $\geq$0.05 is reduced to 1.5% when 95 grains are dated, and if k=117, p is only 0.9%.

As an example that is closer to the worst-case scenario, we now consider a dataset of 155 $^{40}$Ar/$^{39}$Ar ages on lunar spherules collected by Apollo 14 [12]. The age-histogram of these data is more evenly distributed than was the case for the Nubian Sandstone (Figure 7). The maximum probability for missing at least one fraction $\geq$0.05 when only 60 grains are dated is 28%. When 95 grains are dated, the probability of missing at least one fraction $\geq$0.05 is reduced to 3.7%. Finally, dating 117 grains results in p=1.2%.